Combinatorial manifolds with complementarity

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Abstract

A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.

We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d=0, 2, 4, 8, or 16 with n=3d/2+3 and |M| is a “manifold like a projective plane”. Arnoux and Marin had earlier proved the converse statement.